Ten Dice and Divisibility by Seven

You have ten independent fair six-sided dice.

Roll all ten and let $S$ be the sum.
• If $S$ is a multiple of 7, you receive $\$7$.
• Otherwise you receive nothing.
• The game costs $\$1$ to play.

Subproblem A

Should a risk-neutral player play this game?

Subproblem B

Instead of computing the probabilities of every exact possible sum, solve the problem by tracking only information about the running sum modulo 7.

In particular, reduce the state description to whether the current sum is a multiple of 7 or is not a multiple of 7.